We study the question of local and global uniqueness of completions,
based on null geodesics, of Lorentzian manifolds. We
show local uniqueness of such boundary extensions. We give a
necessary and sufficient condition for existence of unique maximal
completions. The condition is verified in several situations of interest.
This leads to existence and uniqueness of maximal spacelike
conformal boundaries, of maximal strongly causal boundaries, as
well as uniqueness of conformal boundary extensions for asymptotically
simple space-times. Examples of applications include the
definition of mass, or the classification of inequivalent extensions
across a Cauchy horizon of the Taub space-time.